Two-point estimates in
probabilities
Emilio Rosenblueth
Universidad Nacional A utfnoma de Mdxico 20, D.F., Mexico
(Received April 1981)
Introduction
In a multitude of problems uncertainty in data and in
theories is significant to such an extent that a probabilistic
treatment ought to be mandatory. Frequently though a
deterministic treatment is preferred so as to remove the
complications of a rigorous probabilistic analysis. Random
variables are then replaced with point estimates, that is,
each variable is replaced with a central value (expectation,
median, or mode), or with one consciously biased so as to
incur errors in the less unfavourable sign, and the estimates
are treated as deterministic. Results are also expressed as
point estimates without even giving an idea of their differences with the corresponding central values or of the
magnitude of the resulting bias or dispersion. In decision
making a trustworthy calculation of the first moment expectation - of functions of the random variables would
often suffice.
This paper develops a simple procedure for computing
the first three moments which overcome the deficiencies of
deterministic treatment by sacrificing the accuracy of a
rigorous probabilistic analysis; the procedure also furnishes
an approximate and an equally simple approach to Bayesian
statistics. For functions of a single variable we propose to
estimate the variable and its functions at two points, rather
than at a single point as in the deterministic approach. We
arrive at the estimate at 2 n points for functions o f n variables although the number of estimates can be reduced to
2n in certain cases of great practical interest. Exceptionally
we would suggest the use of estimates at a larger number of
points to improve the accuracy. In short, the method
presented here gives results that are usually almost as satisfactory as those of a rigorous probabilistic treatment provided the coefficients of variation of independent variables
do not exceed moderate limits, yet it implies no more than
a modest increase in numerical complexity over that of a
purely deterministic analysis.
The method is expounded in a deficient and excessively
succinct manner elsewhere, t'2 which is overcome in the
present paper.
Let Y = Y(X), X = random variable. When we are not
interested in the distribution of Y but only in an approximation to its first few moments we can ignore X's probability density function and use no more than the corresponding moments. The solution will be independent of the
distribution we assign X. Any distribution we assign it
having the same first moments as the given distribution will
fiirnish the exact solution when Y is a linear function of X,
and we can choose X's distribution so that the solution be
sufficiently accurate if Y(X) is sufficiently smooth in the
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© IPC Business Press
neighbourhood of the expectation of X provided X's
dispersion is not too large. We will choose the fictitious
distribution of X following this criterion and that of
simplicity. If we are interested in only Y's expectation we
can assume that the density of X is completely concentrated
at X's expected value. The analysis will correspond to a
point estimate of Y where X equals the expectation of X.
It will be as simple as a deterministic analysis with the
advantage that the meaning of the values computed will be
explicit. We will obtain a first-order approximation to the
expectation of Y which, however, will often be excessively
crude.
If instead of one we use two concentrations, equal to 0.5
each, placed symmetrically with respect to X's expectation,
we can take into account X's first two moments, estimate
those of Y with a second-order approximation to its expected value. If we omit the condition of symmetry we
have enough parameters to take into account the first three
moments of X and obtain a third-order approximation to
the expectation of Y without thereby complicating the
analysis beyond a two-point estimate of Y. In the text we
will adopt the latter approach and derive the results of
concentrations symmetric with respect to the expectation
of X as a particular case.
F u n c t i o n o f one variable
Consider a real function Y = Y(X) of the real random
variable X. (Capitals are used for random variables; the
corresponding lower-case letters denote specific values of
those variables,) Of the latter we know the first three
moments. The ith moment is defined as:
~[i(X) = _[ xipx(x) dx
i = 1, 2
__oo
px(x) is the probability density function of X at X = x.
Mternatively w.~ can write the central moments:
oo
M[.(X) = f (x - .~)ipx(x) dx
i = 1, 2 . . . .
(l)
27 denotes the expectation o f X , MI(X). It follows from
equation (1) that the zero order moment is
oo
f px(X)dx
Appl. Math. Modelling, 1981, Vol. 5, October 1981
329
Two-point estimates in probabilities: E. Rosenblueth
This is always l . The first central moment, M~ (X), is'always
nil. M j ( X ) is k n o w n as the variance o f X and is ordinarily
designated o2; its square root, o, is X ' s standard deviation.
M ~ ( X ) is k n o w n as the skewness o f X ; i f we denote it as
vo 3, v is X ' s skewness coefficient. For symmetric distributions skewness is nil and the same holds for all other central
n
0
1
2
5
10
li
moments of odd order.
We are interested in obtaining expressions for the
exlSectation, standard deviation, and coefficient of skewness
of Y, respectively Y, ¢ry, and v~, andto do it independently
o f X's distribution. To this end let us assign X an arbitrary
distribution having four parameters so as to comply with
expressions for the moment of order zero and for the first
three moments ofX. A particularly simple function satisfying this requirement consists in two concentrations, Pt and
P2, o f the probability density function px(x), respectively
a t X =xz and x2:
t
/n:O
\
\
~x/n = 1
\
,n=2
\\
\
0
0
0 = ~lPt - ~2P2
Curve
1
] =
~Pt
v =
~3tPl -- ~3P2
~e~
2
3
4
6
6
is:
G = v/2 + x / ~ (v/2) 2
(2)
~2 =
(3)
P,
~l -
1
x
Figure 2 Densities of formPx(x) = (n + 1)(1 --x} n
=PI +P2
Their solution
v
0
0.565
0.8607
1.2831
1.5524
n =1 0 ~
5(') is Dirac's delta function of variable (-). The distribution is depicted schematically in Figure 1.
If we make ~i = ]xi - XI]o, i = 1, 2, the four expressions
that are of interest become:
+
V
0.5774
0.7071
0.7746
0.8660
0.9199
'
\
p x ( x ) = e t 6 ( x - x O + P26(x - x2)
1
?
0.5O00
0.3333
0.2500
0.1429
0.0833
v
0.3333
0.4000
0.4867
0.5333
0.6000
0.6667
V
v
0.7071 0.5657
0.5401 04781
0.4403 0.1913
0.3853 -Q 1913
0.3800-O.4761
0.3536 -0. 5667
~2
= ~
P~ =
l
(4)
-e,
(5)
With Yi = y(xi) the first three central moments of Y are:
0 =(Yl-
Y)PI +(Y2- Y)P2
(6)
o=y= (Yt - Y)~P, + (Y2 - Y)2P2
(7)
VyO~ = 0 ' , -- Y)~Pt + (Y2 -- Y)~?=
(8)
From equations (5)-(8) we find:
F "-P~y~ +P2Y2
(9)
Oy":" Px/'P~tP2IYl-Y2I
(10)
V v O r - (e2 - e~)(Yt -Y~)
(I1)
( - means equal to, except for higher-order terms.)
o
x
Figure3 Triangular distributions
J
°
,
-t---4
£20
Et 0
X
Figure I Concentrations of probability density function
330
Appl. Math. Modelling, 1981, Vol. 5, October 1981
Results for Y are not very sensitive to v and computation
of this parameter can be awkward. To facilitate its estimation as well as that of the coefficient of variation F = a/)~
through mere inspection, representative curves are given in
Figures 2-4.
1.O
0.8
0.6
0.4
0.2
0
!
i
Two-point estimates in probabilities: E. Rosenblueth
Curve
1
2
3
4
5
V
1.000
0707
0577
1,311
0472
0.100
6
of which the appropriate smoothness conditions are met,
and apply to each segment, the method proposed. This is
especially useful for computing Y, as computation of the
global o r and t,r is awkward and it may be preferable to
obtain them through analytical or numerical integration.
v
2O00
1.414
1.155
6.185
1.623
0303
F u n c t i o n o f several v a r i a b l e s
1
1
2
b
3
x
4
5
10 x 10 6
Q.x 5~Id e
~6
0
5OO0OO
100O00O
I
1500 0 0 0
x
Figure 4
(a) Gamma and (b) Iognormal distributions
When one of the central moments of X is unknown the
number of simultaneous equaiions is reduced to three, so
we can arbitrarily assign the value of one of the four parameters of the distribution of X. On the other hand, when
t, = 0 the expressions obtained become simpler. They
become:
~l -- ~2 = 1
(12)
Px = P2 = 1/2
(13)
Generalization o f the foregoing method to functions of
several variables requires solution of large numbers of
simultaneous equations, many of which are generally nonlinear (see Appendix). For example, if we know the
moments of the first three orders of the random variables
and there are two such variables, we must solve 10 simultaneous equations, some of which are linear, others quadratic, and the res.t cubic. As unknowns we may choose the
coordinates of four points in the plane of the random
variables Xx, X2, as well as the magnitudes of the concentrations, at these points, of the joint probability function
Px, x2(Xl, x2), which yields 12 unknowns. This number
exceeds that of the equations, so we may arbitrarily choose
the values of two variables or of two relations between
them. With thee random variables the number of simultaneous equations rises to 20 and we may take as unknowns
the coordinates of five points and the corresponding
concentrations.
This approach is awkward. We prefer to concentrate the
density function at a superabundant number of points and
impose conditions on their coordinates. If we take 2 n points
when the number of random variables is n and we leave as
unknowns the concentrations at all points and the coordinates of two of them not having coordinates in common,
distributing the rest so as to form a rectangle, prism, or
hyperprism, we obtain an adequate number of unknowns
to satisfy the moments of orders zero, first, and second of
the form
oo
f (xt
x,) 2
pxi(m) dxi
--oo
i = 1. . . . , n, pxi(xi) marginal probability density function
o f x i. We thus, however, force the other third order
moments without necessarily satisfying the corresponding
conditions, but the sacrifice implies a significant simplification. The resulting equations are simple and can indeed be
solved almost by inspection.
=
Y - (112)(yl +Y2)
(14)
o r - - (I/2) ly~ -Y21
(15)
whence:
Vr "- lYl - Y2I/(Yl +Y2)
(16)
These equations are valid whether v is unknown, zero, or is
regarded as negligible.
Compared with the results of expandingy/, i = I, 2 . . . . .
in a Taylor series about X, multiplying both members by
px(x), and integrating, which furnishes moment M i ( Y ) , we
find that when the third derivative of Y(X) exists equation
(9) constitutes a third order approximation (with relative
errors of the order of V4), equations (10) and (14) are of
second order (with relative errors of the order of Va), and
equations (11), (15), and (16) of first order (with relative
errors of the order of V2). However, the expressions
obtained do not presuppose conditions of continuity or of
existence of the derivatives of Y(X), although the accuracy
of the approximations can seriously deteriorate when Y(X)
is not sufficiently smooth. If Y(X) or its first derivatives
have no more than a f'mite number of discontinuities and
these are rmite, we can divide X into segments within each
O=
4 [ ( 1 * ( V 1 1 2 ) 3 ] [ ( l * ( v 2 / 2 ) 3]
En 0.1
P,2% -o
Pn P21"O
"_~210"2
C
E120
_____#
I
w~2~2.oL_ ~
I
'P, P22-0
II
" E120.! I
xI
Figure 5 Concentrations f o r a f u n c t i o n o f t w o variables
Appl. Math. Modelling, 1981, Vol. 5, October 1981 331
Two-point estimates in probabilities: E. Rosenblueth
01
and P(A) = P(A IX = x 0 Pl + P(A IX = x2) P2 = previous
probability o f A .
Calculation o f x t , x 2 , P~, and P2 proceeds as for functions
o f a single random variable. Once P'l and P2 have been
obtained we can compute the probability o f other experimental results. Generalization to more than one random
variable presents no difficulties.
Occasionally the 'experiment' consists o f a long chain of
elementary experiments, all yielding the same result in an
almost systematic way and its growth makes px(x) tend to
values outside the interval between xz and x2. To cover
this possibility it may be advisable to concentrate the probabilities at more than two points, placing the end points
sufficiently far from X so that the values to which px(x)
can tend will fall, with near certainty, between them.
_1
!(_~)/4
(1-p)/4
(1op)14 i_
(1-p)/4
LI
X~
Figure 6 Special case v, = ~2 = 0
Thus for the case Y = Y(Xb X2) we obtain the rectangle
in Figure 5, where p = coefficient of variation of X1 and X2;
in symbols Pii and ~ii the first subscript indicates the
variable and the second one identifies each o f the values
that the variable can assume;Pii and ai are computed as for
functions o f a single variable. When X~ and X2 are not
correlated, p = 0. If the variables are statistically independent, the solution in Figzlre 5 is correct even for all the
third-order moments, since:
Examples
Smooth function of a single random variable
Consider the following cases: Y = X a; X = 1; o = 0.1, 0.2,
0.5; v = 0, 0.2, 0.6, 0.8. Results are given in Table 1. We
will illustrate their computation for the case a = 0.5, v = 0.4.
From equation (2):
~1 = 0.2 + ~/1 + 0.2 ~ = 1.2198
oo
= 1 + 1.2198 x 0.5 = 1.6099
dxj = 0,
From equation (3):
i , / = 1,2
~2 = 1.2198 - 0.4 = 0.8198
In the special case vl = v2 = O, Figure 5 becomes Figure 6.
The case in which Yis a product o f functions, each a
function o f one o f the random variables, and the latter are
statistically independent is particularly interesting. The
following relations are then exact:
Y = Y, Y2... Yn
(17)
x2 = 1 - 0.8198 x 0.5 = 0.5901
According to equation (4):
P1 =
= 0.4019
P2 = 1 - 0.4019 = 0.5981
Applying equations (9)-(11) we obtain:
x(1 +3V~+v2V])...
Y - 0.4019 x 1.60993 + 0.5981 x 0.59013 = 1.8000
x (l + 3V2n+ vnV3n),
vi = vri
1.2198 + 0 . 8 1 9 8
and following equation (5):
I + V~-= (I + Vx2)(l + V ~ ) . . . (I + Vn~), Vi = Vy/ (18)
1 + 3v~. + ~ r v ~ . = (1 + 3 v =, + ~,v~)
0.8198
or- - ~/0.4019 x 0.598111.60993 - 0.590131 = 1.9450
(19)
v r - (1/1.9450)(0.5981 - 0.4019)(1.60993 - 0.59013)
Each function Yi may consequently be treated separately
and results combined in accordance with equations (17)(19).
= 0.400
Table 1 Statistics of Y, for a smooth function of a single random
variable
B a y e s i a n statistics
p
Let X be a real random variable whose probability density
function we replace with concentrations P~ and P2 placed
respectively at the points whose ordinates are X, and X2
(Figure 1). Suppose that an experiment (or observation) is
performed whose result we designate by A. This will have
modified probabilities P~ and P2 as follows, according to
Bayes' theorem or formula of the probabilities of hypotheses:
0
e; = e(A IX = xt) e~
i = 1, 2
Pt' = value o f P i in the light o f result A, P(A I X = xi) = probability that we obtain result A given that X equalled xi,
332
Appl. Math. Modelling, 1981, Vol. 5, October 1981
a=0.2
a=0.5
1.7500
~/
1.0300
1.1200
ay
0.3010
0.6080
1.6250
vy
0
0
0
0.2
Y
ay
vy
1.0301
0.3071
0.2002
1.1215
0.6323
0.2002
1.7748
1.7800
0.2002
0.4
Y
ay
1.0304
0.3132
1.1232
0.6573
1.8000
vy
0.4000
0.4000
~(
ay
uy
1.0271
0.3201
0.8000
1.1169
1.8076
0.6952
0.8000
2.2424
0.8000
(20)
P(A)
a=0.1
0.8
1.9450
0.4000
Two-point estimates in probabilities: E. Rosenblueth
to assign X a prior distribution uniform between 0 and I :
a
1-
X
o
o5
0.5625
1
1.5
O.75 = 5 ~ y 562
2
o
px'~x) = 1, 0 <~x ~< 1. The object is tossed n times, n = O, 1,
o~"
/
-x-1
I
2
1
x
1--
b
= - -
o
1
-
x
2
x
Figure 7 Data for second problem. (a), function ofx with dis-
2, . . . and in all o f them face exposed is a tail. It is desired
to know the posterior distribution o f X in the light o f this
eventuality.
For the prior distribution o f X (n = 0), we c o m p u t e
X = 0.5, o = ~ / i 2 = 0.2887, v = 0, ~l = ~2 = 1 , x l = 0.7887,
x2 = 0.2113, P~ =/)2 = 0.5. After one toss we fred P~ = 0.2113,
P2 = 0.7887, )7 = 2 x 0.2113 x 0.7887 = 0.3333, etc. Results
are given in Table 2, where t h e y are compared with the
exact solution. The approximation is satisfactory for
small n.
Although a very long series o f tails is a priori unlikely,
ordinates XI and X= make the Bayesian t r e a t m e n t incapable o f adequately representing the results o f a long
series o f tosses giving heads and tails in such p r o p o r t i o n
that X tends asymptotically to any value appreciably higher
than X~ or smaller than X2, which would n o t be too strange.
To remedy this situation we will choose four points to concentrate the probability density function o f X: x = 1, 0.75,
0.25, and 0. F r o m s y m m e t r y , PI = P4 and/)2 =/)3. For the
prior distribution, P1 = P= = 0.5 and 0.52P1 + 0.252P2 = o2/2
= 1/24. Hence PI = 1/18 a n d P 2 = 4/9. For n >~ 1, equation
continuous derivative; (b), probability density o f x .
We notice in Table I that Y is only slightly sensitive to ~,
o r- is more so, and computation o f Vr is pointless i f v is
erroneously taken as zero.
Pa
P.
P22
P~2
3
Discontinuity in the first derivative
2.6099
L e t y = 0 f o r x ~< 1 , y = x - 1 for l < x <<,2, p x ( x ) =
( 3 / 4 ) ( 2 - x ) x (Figure 7). We will be content to c o m p u t e
Y. We directly obtain ,Y = 1, o = 0.4472, v = 0.5. If we
ignored the break a t x = 1 we would find ~1 = ~2 = 1,
x~ = 1.4472,x= = 0.5528,P~ =P2 = 0.5,y~ = 0.4472,y~ = 0,
and Y - 0.2236. In contrast, proceeding b y segments, for
x / > 1 we obtain ,Y = 1.375, o = 0.6778, ~, = 0.4015,
~ = 1.2207, ~2 = 0.8192, P~ = 0.2008, P= = 0.2992 (notice
that P~ and P : are half o f the values yielded b y equations
(4) and (5); this is due to M0 being 0.5 for the segment
considered), x~ = 2.2024, x=--- 0.8198 and, extrapolating
the f u n c t i o n y = x - 1, we f'md y~ = 1.2024,y= = - 0.1802.
In the segment x ~< 1, y~ and y= are nil. According to
equation (9), Y = 0.1875. This result coincides with the
exact answer because in each segment Y(X) is linear and
px(x) is quadratic.
x2--2
1.5901
1 -
U
Function o f two variables
Consider the example Y = XiX2,-~l
a a
= 1,-~2 = 2, o 1 = 0 . 2 ,
o2 = 0.5, t'l = 0.4, p = 0.4. Using equations (2)-(5) we obtain
~11 = 1.1050, ~12 = 0 . 9 0 5 0 , x l t = 1.2210,x12 = 0.8190,
P11 = 0.4502,P12 = 0.5498, ~2t = 1.2198, ~22 = 0.8198,
x21 = 2.6099,x22 = 1.5091 ,P=1 = 0.4019,P22 = 0.5981.
According to Figure 5 we compute the concentrations
shown in Figure 8. Thence Y = 12.4173, E Y 2 = 314.7407,
so that o ~ = E Y 2 - ~,2= 160.5520, or- = 12.6709. Compare
with the result o f assuming v~ = r2 = O. Using results o f
example 1 we get Y = 12.1296, or-= 11.1252.
Bayesian statistics
An irregularly-shaped object has two faces, which we
will designate respectively 'head' and 'tail'. A priori we do
n o t know the probability X that on tossing the object the
head be uppermost. In view o f this ignorance it is decided
PI1
P12
P21
P22
I
0.8190 X = I
0
=
=
=
=
O. 2785
0.1717
0.1234
O. 4264
I
1.2210
Xl
Figure 8 C o n c e n t r a t i o n s f o r t h i r d e x a m p l e
Table 2 Probability that in toss n + 1 we obtain a head when the
first n have come.out tails
n
(1)
(2)
(3)
0
1
2
3
4
5
0.5000
0.3333
0.2500
0.2222
0.2143
0.21 21
0.5000
0.3333
0.2500
0.2083
0.1842
0.1653
O.OOOO
0.5000
0.3333
0.2500
0.2000
0.1667
0.1429
0.0000
0.2113
(1) with two concentrations of PX(X)
(2) with four concentrations o f P x ( x )
(3) exact answer
A p p l . M a t h . M o d e l l i n g , 1 9 8 1 , V o l . 5, O c t o b e r 1981
333
T w o - p o i n t estimates in p r o b a b i l i t i e s : E. R o s e n b l u e t h
(20) gives:
e;= y ei4
i
whence:
Y eixT(l -xi)
i
i
The method propounded lends itself to application in
Bayesian statistics. It suffices to replace the prior probability
density function of the variables of interest with concentrations in the usual way and to modify the magnitudes of the
concentrations as a function of the statistical information,
applying Bayes' theorem to compute their posterior values.
In some cases it is advisable to introduce a larger number of
concentrations to cover the possibility that statistical data
make the probability density function evolve markedly
toward values outside the range de£med by the usual
concentrations.
Substituting numerical values we obtain:
X=
( 3 1 1 6 ) ( I + 3 " - ' ) 1 4 n-'
118 + ( I + 3n)14 n
Acknowledgement
T h u s w e o b t a i n the results given in Table 2. I t s h o u l d be
The author thanks L. Esteva for his valuable suggestions and
constructive criticism of this paper.
noted that errors for moderate and large n decrease
perceptibly
References
S u m m a r y and c o n c l u s i o n s
Many practical problems would require a probabilistic treatment that is generally not carried out because it would be
too time consuming if done rigorously. In this paper an
approximate method is proposed, which is very simple and
only sacrifices the accuracy slightly provided that the
dispersions of the variables are not too large. The method
allows estimating the first three moments of a function of
random variables of which the first three moments are
known. This is especially useful in decision theory, in which
it is enough to estimate the first moment, or expectation,
of the dependent variable.
For functions Y of a single random variable X the probability density function of X is replaced by two concentrations. Expressions are available which furnish the magnitudes and ordinates of the concentrations in terms of X's
first three moments, and thence it is straightforward to
compute approximations to the corresponding moments
of Y. When Yis sufficiently smooth the approximation to
Y's expectation is third order; the one to its standard deviation and coefficient of variation are second order; and first
order for Y's skewness and skewness coefficient. When Y or
its first derivative have t'mite discontinuities the same
expressions can still be used though possibly with too great
a loss of accuracy. The situation is overcome by applying
the proposed method to each of a series of segments.
If Y is a function of two or more variables a larger
number of concentrations are necessary. Their magnitudes
and coordinates can be computed by solving certain simultaneous equations so as to satisfy the first moments of the
variables, but the number of such equations increases
rapidly with the number of variables, especially when the
latter are not statistically independent. It is found preferable to resort to a superabundant number of concentrations
located at the vertices of a rectangle, prism, or hyperprism.
Solution of the simultaneous equations can then be obtained
almost by inspection. With this artifice the solution is particularly simple when the tlfird moments are zero or are assumed
to be zero. In a frequent practical case Y is the product of
functions each of a single random variable and these variables
are statistically independent. Two concentrations per
variable suffice then and expressions are available for the first
three moments of Y which are exact in terms of the corresponding moments of the functions.
334
A p p l . M a t h . M o d e l l i n g , 1981, V o l . 5, O c t o b e r 1981
1 Rosenblueth, E. 'Aproximaciones de segundos momentos en
probabilidades', Boletbz del blstituto Mexicano de Planeacidn y
Operacidn de Sistemas 1974, 26, 1
2 Rosenblueth, E. Proc. Nat. Acad. Sci, USA 1975, 72, (10), 3812
3 Benjamin, J. R. and Cornell, C. A. 'Probability statistics, and
decision for civil engineers', McGraw-Hill,New York, 1970
Appendix
C o n d i t i o n s to satisfy, c o n c e n t r a t i o n s , a n d n u m b e r
of redundaut parameters
Let n be the number of random variables, i the order of the
moments of p, and N i the number of moments of order i
to be satisfied. Then the number of conditions to be met,
and hence the number of imposed equations, for all
moments of p, from the one of order 0 to those of order k,
Table A 1 Number of conditions to be met
n
1
2
3
4
5
6
7
No
No+N t
No+N~+N =
No+Nz+N=+N ~
1
2
3
4
1
3
6
10
1
4
10
20
1
5
15
35
1
6
21
56
1
7
28
84
1
8
36
120
Table A2 Number of concentrations and number of redundant
parameters
(1)
(2)
(3)
(4)
(5)
(6)
1
2
3
4
5
6
7
4
10
20
35
56
84
120
2
3.3
5
7
9.3
12
15
2
4
5
7
10
12
15
0
2
0
0
4
0
0
0
2
12
45
136
364
904
(1)n
(2) N o + N I + N 2 + N 3 = number of equations imposed by number of
moments o f p
(3} ( N 0 + N t + N a + N 3 ) / ( n + 1) = lower bound to number of points
wherep is to be concentrated
(4) smallest number of concentrations o f p
(5) number of redundant parameters
(6) ditto when using 2n concentrations o f p
Two-point estimates in probabilities: E. Rosenblueth
k
is
gi
E (,z + 1)
i=o
k
i=O
and it is a function o f k . We can prove that:
k
(k+n)!
i=o-
k!n!
y g,---
Table A 1 contains the values of this quantity for n between
1 and 7 and k between 0 and 3.
For every point where the probability density function
p is assumed concentrated we can write as many equations as
the point has coordinates, that is n, plus one, the latter
coming from the magnitude of the concentration. The
number of parameters to determine is thus n + 1 times the
number of concentrations. Hence, the number of points
where p is to be concentrated should not be smaller than
and the minimum number of concentrations is the first
integer not smaller than this quantity. This number, times
n + 1, minus the number of equations imposed gives the
number of redundant parameters resulting from taking the
smallest possible number of concentrations o f p .
On the other hand if we use 2 n concentrations of the
probability density functions we obtain
k
( . + 0 2 " - Y. g~
i=0
redundant parameters.
The results of these conditions for k = 3 are given in
Table A 2 .
Appl. Math. Modelling, 1981, Vol. 5, October 1981 335